# Definition:General Logarithm/Common

## Contents

## Definition

Logarithms base $10$ are often referred to as common logarithms.

## Examples

### Common Logarithm: $\log_{10} 2$

The common logarithm of $2$ is:

- $\log_{10} 2 \approx 0.30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$

This sequence is A007524 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

### Common Logarithm: $\log_{10} 3$

The common logarithm of $3$ is:

- $\log_{10} 3 = 0.47712 \, 12547 \, 19662 \, 43729 \, 50279 \ldots$

This sequence is A114490 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

### Common Logarithm: $\log_{10} e$

The common logarithm of Euler's number $e$ is:

- $\log_{10} e = 0 \cdotp 43429 \, 44819 \, 03251 \, 82765 \, 11289 \, 18916 \, 60508 \, 22943 \, 97005 \, 803 \ldots$

This sequence is A002285 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

### Common Logarithm: $\log_{10} \pi$

The common logarithm of $\pi$ is:

- $\log_{10} \pi = 0.49714 \, 98726 \, 94133 \, 85435 \, 12683 \ldots$

This sequence is A053511 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Also known as

**Common logarithms** are sometimes referred to as **Briggsian logarithms** or **Briggs's logarithms**, for Henry Briggs.

In elementary textbooks and on most pocket calculators, $\log$ is assumed to mean $\log_{10}$.

This ambiguous notation is not recommended, particularly since $\log$ often means base $e$ in more advanced textbooks.

## Also see

- Results about
**logarithms**can be found here.

## Historical Note

**Common logarithms** were developed by Henry Briggs, as a direct offshoot of the work of John Napier.

After seeing the tables that Napier published, Briggs consulted Napier, and suggested defining them differently, using base $10$.

In $1617$, Briggs published a set of tables of logarithms of the first $1000$ positive integers.

In $1624$, he published tables of logarithms which included $30 \, 000$ logarithms going up to $14$ decimal places.

Before the advent of cheap means of electronic calculation, **common logarithms** were widely used as a technique for performing multiplication.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 7$: Common Logarithms and Antilogarithms - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $10$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $10$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Briggsian logarithm** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $5$: Eternal Triangles